EH is a diameter of D. The measure of is (10x + 8)° and the measure of is (11x)°. Determine the values. The value of x is . The measure of is degrees. The measure of is degrees.

The value of 'x' is 5, the measure of arc EF is 58 degrees, and the measure of arc DH is 55 degrees and this can be determined by using the given data.

Given :

EH is the diameter of circle D.
The measure of arc EF is (10x + 8)° and the measure of arc GH is (11x)°.
Lines segments FD and GD are radii.
Angle FDG is 67 degrees.

The following steps can be used in order to determine the measure of arc EF and arc GH:

Step 1 - According to the given data, EH is the diameter of circle D. So, the angle EDH is:

[tex]\angle EDF+\angle GDH+\angle FDG = \angle EDH[/tex]

Step 2 - Now, substitute the values of all the angles in the above expression.

[tex](10x+8)+(11x)+67 = 180[/tex]

Step 3 - Simplify the above expression.

[tex]\begin{aligned}\\21x &= 105\\x &= 5\\\end{aligned}[/tex]

Step 4 - Now, the measure of arc EF is:

[tex]\angle EDF = 10(5)+8[/tex]

[tex]\angle EDF = 58^\circ[/tex]

Step 5 - The measure of arc GH is:

[tex]\angle GDH = 11\times 5[/tex]

[tex]\angle GDH= 55^\circ[/tex]

The value of 'x' is 5, the measure of arc EF is 58 degrees, and the measure of arc DH is 55 degrees.

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